For the pair of functions defined, find (ƒ+g)(x), (ƒ-g)(x), (ƒg)(x), and (f/g)(x).Give the domain of each. See Example 2. ƒ(x)=√(5x-4), g(x)=-(1/x)

Verified step by step guidance

1. Step 1: To find \((f+g)(x)\), add the functions: \(f(x) + g(x) = \sqrt{5x-4} - \frac{1}{x}\).
2. Step 2: Determine the domain of \((f+g)(x)\). The domain is restricted by both \(f(x)\) and \(g(x)\). For \(f(x) = \sqrt{5x-4}\), the expression under the square root must be non-negative, so \(5x-4 \geq 0\). Solve for \(x\) to find \(x \geq \frac{4}{5}\). For \(g(x) = -\frac{1}{x}\), \(x\) cannot be zero. Combine these restrictions to find the domain.
3. Step 3: To find \((f-g)(x)\), subtract the functions: \(f(x) - g(x) = \sqrt{5x-4} + \frac{1}{x}\).
4. Step 4: The domain of \((f-g)(x)\) is the same as \((f+g)(x)\) since the restrictions on \(x\) are identical.
5. Step 5: To find \((fg)(x)\) and \((f/g)(x)\), multiply and divide the functions respectively: \((fg)(x) = \sqrt{5x-4} \cdot \left(-\frac{1}{x}\right)\) and \((f/g)(x) = \frac{\sqrt{5x-4}}{-\frac{1}{x}}\). Determine the domain for each by considering the restrictions from both functions and ensuring the denominator is not zero for \((f/g)(x)\).

Key Concepts

Function Operations

Function operations involve combining two functions through addition, subtraction, multiplication, or division. For example, if ƒ(x) and g(x) are two functions, (ƒ+g)(x) represents the sum of the two functions evaluated at x. Understanding how to perform these operations is essential for manipulating and analyzing functions in algebra.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For example, in the function ƒ(x)=√(5x-4), the expression under the square root must be non-negative, leading to a restriction on x. Similarly, for g(x)=-(1/x), x cannot be zero, as division by zero is undefined. Identifying the domain is crucial for ensuring valid function operations.

Composite Functions

Composite functions are formed when one function is applied to the result of another function. In the context of the given question, understanding how to combine ƒ(x) and g(x) through operations like addition or multiplication is key to finding (ƒ+g)(x), (ƒ-g)(x), (ƒg)(x), and (f/g)(x). This concept helps in analyzing the behavior and characteristics of the resulting functions.